How to observe if a vector field has curl or not?

In summary, the vector fields in the images have curl, but it is not always easy to determine whether or not there is curl.
  • #1
Homework Statement
So I am supposed to find out if any of the vector fields given has zero curl or not.
Relevant Equations
So I know that curl has something to do with rotation, and I know how to calculate this vector, but not how to observe if it is zero or not from a picture of the vector field.
These are the vector fields. I really have no idea how to see if there is a curl or not. I have been looking at the rotation of the vector fields. The fields d and e seem to have some rotation or circular paths, but I read online that curl is not about the rotation of the vector field itself, but more about the rotation of points in the vector field. Can someone help me understand how to determine if there is a curl when one are given a picture of a vector field like the ones below?

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  • #2
Curl equal zero implies that the path integral of a vector field around some closed path is zero. Of course i am supposing that there are no singularities inside this path, as fas as i remember the name of this condition is to have a simply connected domain.

Anyway, try to follow the reasoning below, for example for the first image:
The arrows more to left have a magnitude lesser than the arrows at the center. Also, both arrows points up. Let's try to follow the path i did at the image:
The arrow path has contribuition ##\int F dr##, the black paths have no contribuition, since they are perpendicular. For the green path, ##\int F' dr'##, where ##|F'|>|F|##, note that the contribuition here is negative.

The total integral is ##\int F dr - F' dr \neq 0##. So curl of F is not zero.
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  • #3
Curl can be deceptive if you are trying to "eyeball" it. Keep in mind that parallel flow lines can show curl if they show changing velocities in side-by-side flow lines. Also, flow lines that appear to curve may not show curl if the lengths of the side-by-side lines are equal (as long as the center of the curvature is not in the field of interest). So what would be your answers for the examples you show? Your answer may be one thing for particular points of a diagram and something else for other parts of the diagram.
 
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1. How do I determine if a vector field has curl?

To determine if a vector field has curl, you can use the curl operator, which is represented by the symbol ∇ x. Apply this operator to the vector field and if the resulting value is non-zero, then the vector field has curl.

2. What is the physical significance of a vector field having curl?

A vector field with curl represents the presence of rotation or circulation in the field. This is commonly observed in fluid flow or electromagnetic fields. The magnitude of the curl also indicates the strength of the rotation or circulation.

3. Can a vector field have both curl and divergence?

Yes, a vector field can have both curl and divergence. In fact, the curl and divergence of a vector field are related through the vector identity ∇ x (∇ · F) = 0. This means that if a vector field has non-zero curl, it must also have non-zero divergence and vice versa.

4. How can I visualize the curl of a vector field?

One way to visualize the curl of a vector field is by using streamlines. These are curves that are tangent to the vector field at every point. If the streamlines are closed loops, it indicates the presence of curl. Another way is to use a curlometer, which is a device that measures the magnitude and direction of curl at a specific point in the field.

5. What are some real-world applications of vector fields with curl?

Vector fields with curl have many practical applications. In fluid dynamics, they are used to model the flow of air and water around objects such as airplanes and ships. In electromagnetism, they are used to describe the behavior of magnetic fields and the flow of electric current. They are also used in meteorology to study atmospheric circulation patterns and in geology to model the movement of tectonic plates.

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